In a large saucepan or Dutch oven, saute the first three ingredients in butter until tender. Add the chicken and vegetable broths, barley and rosemary. Bring to a boil. Reduce heat; cover and simmer for 30 minutes or until barley is tender. Add broccoli; cover and cook for 10 minutes or until broccoli is tender.
In a small bowl, combine cornstarch and cold water until smooth; stir into the soup. Bring to a boil; cook and stir for 2 minutes or until thickened. Reduce heat; stir in the cream, salt and pepper (do not boil). Sprinkle with Parmesan cheese.